survey of numerical electromagnetic modeling techniques

7/28/2019 Survey of Numerical Electromagnetic Modeling Techniques

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Title: Survey of Numerical Electromagnetic

Modeling Techniques

ReportNumber: TR91-1-001.3

Author: Dr. Todd H. HubingDept. of Electrical EngineeringUniversity of Missouri-Rolla

Sponsor: Intel Corporation

Date: September 1, 1991

This report reviews and summarizes numerical electromagnetic modeling techniquesthat can be used to analyze sources of electromagnetic interference. Several referencesare cited to aid the reader who wishes to investigate a particular technique in more detail.

University of Missouri-Rolla

Electromagnetic Compatibility Laboratory


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Introduction

Computer techniques have revolutionized the way in which electromagnetic problems are

analyzed. Antenna and microwave engineers rely heavily on computer methods to analyze and help

evaluate new designs or design modifications. Although most EM problems ultimately involve

solving only one or two partial differential equations subject to boundary constraints, very few

practical problems can be solved without the aid of a computer.

Computer methods for analyzing problems in electromagnetics generally fall into one of three

categories, analytical techniques, numerical techniques, and expert systems. Analytical techniques

make simplifying assumptions about the geometry of a problem in order to apply a closed-form (or

table look-up) solution. Numerical techniques attempt to solve fundamental field equations directly,subject to the boundary constraints posed by the geometry. Expert systems do not actually calculate

the field directly, but instead estimate values for the parameters of interest based on a rules database.

A number of computer programs based on analytical techniques are available to the EMC

engineer. Some are very simple and run on personal computers. Others such as IEMCAP [1] are

very elaborate. Analytical techniques can be a useful tool when the important EM interactions of

the configuration can be anticipated. However, most EMC problems of interest are simply too

unpredictable to be modeled using this approach.

Expert systems approach a problem in much the same way as a quick-thinking, experienced EM

engineer with a calculator would approach it. As system design and board layout procedures become

more automated, expert system EM software will certainly play an important role. Nevertheless,

expert systems are no better than their rules database and it is unlikely that they will ever be used

to model or understand the complex EM interactions that cause EMI sources to radiate.

Numerical techniques generally require more computation than analytical techniques or expert

systems, but they are very powerful EM analysis tools. Without making a priori assumptions about

which field interactions are most significant, numerical techniques analyze the entire geometry

provided as input. They calculate the solution to a problem based on afull-wave analysis.

A number of different numerical techniques for solving electromagnetic problems are available.

Each numerical technique is well-suited for the analysis of a particular type of problem. The


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numerical technique used by a particular EM analysis program plays a significant role in determining

what kinds of problems the program will be able to analyze.

This report reviews the strengths and limitations of different numerical techniques for analyzing

electromagnetic configurations. A particular emphasis is placed on how these techniques could beapplied to the analysis of electromagnetic interference (EMI) sources.

EMI Source Models

One problem with attempting to analyze sources of EMI on a computer is that it is difficult to

predict what details of the EMI source need to be modeled. Generally, it is the common-mode

currents in a system that have the biggest impact on the radiated EMI [2,3]. These currents are not

intentionally generated and they cannot be predicted using simple lumped-element circuit-modeling

techniques.

Sources of common-mode current are often difficult to locate. A single printed circuit card may

contain many potential sources and coupling paths. The intentionally generated, differential-mode

currents are generally orders of magnitude larger than the common-mode currents. Because of this,

there has been a tendency for source models to focus on the differential-mode currents. Elaborate

transmission line modeling techniques have been employed to calculate these currents with a high

degree of accuracy. In practical situations however, these currents have been found to have little or

no correlation with the radiated field strengths.

In order to be truly useful, any EMI source modeling technique must be able to model all aspects

of the system that could affect the common-mode current levels. For systems with printed circuit

cards, this means modeling the geometric details of configurations containing metal surfaces, wires,

and dielectrics. In general, these configurations have no particular symmetry and they are neither

electrically large nor small. This presents a significant challenge for a numerical modeling

technique. So far, no one technique has been developed that is able to meet this challenge. As a

result, computer techniques have not been utilized for EMI source modeling to the extent that theyhave for modeling antennas and microwave devices.

The state-of-the-art in numerical electromagnetic modeling is advancing at a rapid pace,

however. Every year hundreds of papers are published describing new techniques, enhancements

to existing techniques, new implementations of existing techniques, and new applications of

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computer modeling. Sorting through this wealth of information in order to choose the computer

technique that is best for a particular application can be overwhelming.

The following sections outline several general numerical modeling techniques that have been

used to analyze EMI source configurations with some success. Each technique is best-suited toanalyze different configurations. No one technique can be used to model all EMI sources, however

each of these techniques can be applied to a number of EMI source configurations. Two or three

of these techniques, collectively, represent a potentially powerful set of tools for the EMI engineer.

Finite Element Methods

Scalar finite element methods are widely used by civil and mechanical engineers to analyze

material and structural problems. Electrical engineers use finite element methods to solve complex,

nonlinear problems in magnetics and electrostatics. Until recently however, very little practical

modeling of 3-dimensional electromagnetic radiation problems was performed using this technique.

There were two reasons for this. First, practical three-dimensional vector problems require

significantly more computation than two-dimensional or scalar problems. Second, spurious solu-

tions known as vector parasites often result in unpredictable, erroneous results. However, recent

developments in this field [4,5] appear to have solved the vector parasite problem. An increasing

availability of computer resources coupled with a desire to model more complex electromagnetic

problems has resulted in a wave of renewed interest in finite element methods for solving EM

radiation problems.

The first step in finite-element analysis is to divide the configuration into a number of small

homogeneous pieces or elements. An example of a finite-element model is shown in Figure 1. The

model contains information about the device geometry, material constants, excitations and boundary

constraints. The elements can be small where geometric details exist and much larger elsewhere.

In each finite element, a simple (often linear) variation of the field quantity is assumed. The corners

of the elements are called nodes. The goal of the finite-element analysis is to determine the field

quantities at the nodes.

Most finite element methods are variational techniques. Variational methods work by minimiz-

ing or maximizing an expression that is known to be stationary about the true solution. Generally,

finite-element analysis techniques solve for the unknown field quantities by minimizing an energy

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functional. The energy functional is an expression describing all the energy associated with the

configuration being analyzed. For 3-dimensional, time-harmonic problems this functional may be

represented as,

F =v

| H | 2

2+

| E | 2

2

JE2j

dv(1)

The first two terms in the integrand represent the energy stored in the magnetic and electric fields

and the third term is the energy dissipated (or supplied) by conduction currents.

Expressing H in terms ofE and setting the derivative of this functional with respect to E equal

to zero, an equation of the formf(J,E)= 0 is obtained. A kth

-order approximation of the functionf

is then applied at each of the N nodes and boundary conditions are enforced, resulting in the system

of equations,

J1J2

Jn

=

y11

y21

y12

y22

ynn

E1E2

En

(2)

The values ofJ on the left-hand side of this equation are referred to as the source terms. They

represent the known excitations. The elements of the Y-matrix are functions of the problem

geometry and boundary constraints. Since each element only interacts with elements in its own

neighborhood, the Y-matrix is generally sparse. The terms of the vector on the right-hand side

Structure Geometry Finite-Element Model

Figure 1: Finite-Element Modeling Example

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represent the unknown electric field at each node. These values are obtained by solving the system

of equations. Other parameters, such as the magnetic field, induced currents, and power loss can

be obtained from the electric field values.

In order to obtain a unique solution, it is necessary to constrain the values of the field at allboundary nodes. For example, the metal box of the model in Figure 1 constrains the tangential

electric field at all boundary nodes to be zero. A major weakness of the finite element method is

that it is relatively difficult to model open configurations (i.e. configurations where the fields are

not known at every point on a closed boundary). Various techniques such as ballooning and

absorbing boundaries are used in practice to overcome this deficiency. These techniques work

reasonably well for 2-dimensional problems, but so far they are not very effective for 3-dimensional

electromagnetic radiation problems.

The major advantage that finite element methods have over other EM modeling techniques stems

from the fact that the electrical and geometric properties of each element can be defined inde-

pendently. This permits the problem to be set up with a large number of small elements in regions

of complex geometry and fewer, larger elements in relatively open regions. Thus it is possible to

model configurations that have complicated geometries and many arbitrarily shaped dielectric

regions in a relatively efficient manner.

Commercial finite element codes [6,7] are available that have graphical user interfaces and can

determine the optimum placement of node points for a given geometry automatically. These codes

are used to model a wide variety of electromagnetic devices such as spark plugs, transformers,

waveguides, and integrated circuits.

Specific implementations of three-dimensional electromagnetic finite element codes are de-

scribed in Ph.D. dissertations by Maile [8] and Webb [9]. Silvester and Ferrari [10] have written

an excellent text on this subject for electrical engineers.

Moment Methods

Like finite-element analysis, the method of moments (or moment method) is a technique for

solving complex integral equations by reducing them to a system of simpler linear equations. In

contrast to the variational approach of the finite element method however, moment methods employ

a technique known as the method of weighted residuals. Actually, the terms method-of-moments

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and method-of-weighted-residuals are synonymous. Harrington [11] was largely responsible for

popularizing the term method of moments in the field of electrical engineering. His pioneering

efforts first demonstrated the power and flexibility of this numerical technique for solving problems

in electromagnetics.

All weighted residual techniques begin by establishing a set of trial solution functions with one

or more variable parameters. The residuals are a measure of the difference between the trial solution

and the true solution. The variable parameters are determined in a manner that guarantees a best fit

of the trial functions based on a minimization of the residuals.

The equation solved by moment method techniques is generally a form of the electric field

integral equation (EFIE) or the magnetic field integral equation (MFIE). Both of these equations

can be derived from Maxwells equations by considering the problem of a field scattered by a perfect

conductor (or a lossless dielectric). These equations are of the form,

EFIE: E=fe(J) (3)

MFIE: H=fm(J) (4)

where the terms on the left-hand side of these equations are incident field quantities and J is the

induced current.

The form of the integral equation used determines which types of problems a moment-method

technique is best suited to solve. For example one form of the EFIE may be particularly well suited

for modeling thin-wire structures, while another form is better suited for analyzing metal plates.

Usually these equations are expressed in the frequency domain, however the method of moments

can also be applied in the time domain.

The first step in the moment-method solution process is to expand J as a finite sum of basis (or

expansion) functions,

J=i=1

M

Jibi

(5)

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where bi is the ith

basis function and Ji is an unknown coefficient. Next, a set of M linearly

independent weighting (or testing) functions, wj, are defined. An inner product of each weighting

function is formed with both sides of the equation being solved. In the case of the MFIE (Equation

4), this results in a set of M independent equations of the form,

= j=1,2,,M (6)

By expandingJusing Equation (5), we obtain a set of M equations in M unknowns,

= j= 1,2,M

(7)

This can be written in matrix form as,

[H]=[Z][J] (8)

where: Zij=

Ji=Ji

Hj=

The vectorHcontains the known incident field quantities and the terms of the Z-matrix are

functions of the geometry. The unknown coefficients of the induced current are the terms of theJ

vector. These values are obtained by solving the system of equations. Other parameters such as

the scattered electric and magnetic fields can be calculated directly from the induced currents.

Depending on the form of the field integral equation used, moment methods can be applied to

configurations of conductors only, homogeneous dielectrics only, or very specific conductor-dielec-

tric geometries. Moment method techniques applied to integral equations are not very effective

when applied to arbitrary configurations with complex geometries or inhomogeneous dielectrics.

They also are not well-suited for analyzing the interior of conductive enclosures or thin plates with

wire attachments on both sides [12].

Nevertheless, moment method techniques do an excellent job of analyzing a wide variety of

important three-dimensional electromagnetic radiation problems. General purpose moment method

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codes are particularly efficient at modeling wire antennas or wires attached to large conductive

surfaces. They are widely used for antenna and electromagnetic scattering analysis. Several

non-commercial general-purpose moment-method computer programs are available [13-17].

Finite Difference Time Domain Method

The Finite Difference Time Domain (FDTD) method is a direct solution of Maxwells time

dependent curl equations,

E=Ht (9)

H=E+E

t (10)

It uses simple central-difference approximations to evaluate the space and time derivatives.

The FDTD method is a time stepping procedure. Inputs are time-sampled analog signals. The

region being modeled is represented by two interleaved grids of discrete points. One grid contains

the points at which the magnetic field is evaluated. The second grid contains the points at which

the electric field is evaluated.

A basic element of the FDTD space lattice is illustrated in Figure 2. Note that each magnetic

field vector component is surrounded by four electric field components. A first-order central-dif-

ference approximation can be expressed as,

1

AEz1(t)+Ey2(t)Ez3(t)Ey4(t)

=

o2t

Hxo(t+t)Hxo(tt) (11)

where A is the area of the near face of the cell in Figure 2. Hxo(t+t) is the only unknown in this

equation, since all other quantities were found in a previous time step. In this way, the electric fieldvalues at time tare used to find the magnetic field values at time t+t. A similar central-difference

approximation of Equation (10) can then be applied to find the electric field values at time t+2t

from the magnetic field values at time t+t. By alternately calculating the electric and magnetic

fields at each time step, fields are propagated throughout the grid.

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Time stepping is continued until a steady state solution or the desired response is obtained. At

each time step, the equations used to update the field components are fully explicit. No system of

linear equations must be solved. The required computer storage and running time is proportional

to the electrical size of the volume being modeled and the grid resolution.

Figure 3 illustrates an arbitrary scatterer embedded in a FDTD space lattice. Special absorbing

elements are used at the outer boundary of the lattice in order to prevent unwanted reflection of

signals that reach this boundary. Values of, and assigned to each field component in each celldefine the position and electrical properties of the scatterer. These parameters can have different

values for different field orientations permitting anisotropic materials to be modeled. Their values

can also be adjusted at each time-step depending on conditions making it easy to model nonlinear

materials.

Because the basic elements are cubes, curved surfaces on a scatterer must be staircased. For

many configurations this does not present a problem. However for configurations with sharp, acute

edges, an adequately staircased approximation may require a very small grid size. This cansignificantly increase the computational size of the problem. Surface conforming FDTD techniques

with non-rectangular elements have been introduced to combat this problem. One of the more

promising of these techniques, which permits each element in the grid to have an arbitrary shape,

is referred to as the Finite Volume Time Domain (FVTD) method [19].

E

E

E

EE

E

E

E

E

z 1

y 2 z 3

y 4

x 1

x 2

x 3

y 1

z 4

H

H

Hx o y o

z oz

x

y

Figure 2: Basic Element of the FDTD Space Lattice

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Frequency domain results can be obtained by applying a discrete Fourier transform to the time

domain results. This requires additional computation, but a wide-band frequency-domain analysis

can be obtained by transforming the systems impulse response.

The FDTD and FVTD methods are widely used for radar cross section analysis although they

have been applied to a wide range of EM modeling problems. Their primary advantage is their

great flexibility. Arbitrary signal waveforms can be modeled as they propagate through complex

configurations of conductors, dielectrics, and lossy non-linear non-isotropic materials. Another

advantage of these techniques is that they are readily implemented on massively parallel computers,

particularly vector processors and SIMD (single-instruction-multiple-data) machines.

Wavetracer Inc. [19] sells a massively parallel computer and FDTD software for EM modeling.

Lawrence Livermore [20] and Boston University [21] have run FDTD algorithms on a Connection

Machine. These implementations have been used to provide animated graphical representations of

EM waves as they propagate in and around a variety of interesting configurations.

The only significant disadvantage of this technique, is that the problem size can easily get out

of hand for some configurations. Thefineness of the grid is generally determined by the dimensions

of the smallest features that need to be modeled. The volume of the grid must be great enough to

encompass the entire object and most of the near field. Large objects with regions that contain

x

y

z

unit

cell

scatterer

Figure 3: Scatterer in an FDTD Space Lattice

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small, complex geometries may require large, dense grids. When this is the case, other numerical

techniques may be much more efficient than the FDTD or FVTD methods.

Finite Difference Frequency Domain Method

Although conceptually the Finite Difference Frequency Domain (FDFD) method is similar to

the Finite Difference Time Domain (FDTD) method, from a practical standpoint it is more closely

related to the finite element method. Like FDTD, this technique results from a finite difference

approximation of Maxwells curl equations. However, in this case the time-harmonic versions of

these equations are employed,

E=jH (12)

H=(+j)E (13)

Since, there is no time stepping it is not necessary to keep the mesh spacing uniform. Therefore

optimal FDFD meshes generally resemble optimal finite element meshes. Like the moment-method

and finite-element techniques, the FDFD technique generates a system of linear equations. The

corresponding matrix is sparse like that of the finite element method.

Although it is conceptually much simpler than the finite element method, very little attention

has been devoted to this technique in the literature. Perhaps this is due to the head startthat finiteelement techniques achieved in the field of structural mechanics.

There are apparently very few codes available that utilize this technique. A notable exception

is the FDFD module that is included in the GEMACS software marketed by Advanced Electromag-

netics [22].

Transmission Line Matrix Method

The Transmission Line Matrix (TLM) method is similar to the FDTD method in terms of its

capabilities, but its approach is unique. Like FDTD, analysis is performed in the time domain and

the entire region of the analysis is gridded. Instead of interleaving E-field and H-field grids however,

a single grid is established and the nodes of this grid are interconnected by virtual transmission

lines. Excitations at the source nodes propagate to adjacent nodes through these transmission lines

at each time step.

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The symmetrical condensed node formulation introduced by Johns [23] has become the standard

for three-dimensional TLM analysis. The basic structure of the symmetrical condensed node is

illustrated in Figure 4. Each node is connected to its neighboring nodes by a pair of orthogonally

polarized transmission lines. Generally, dielectric loading is accomplished by loading nodes with

reactive stubs. These stubs are usually half the length of the mesh spacing and have a characteristic

impedance appropriate for the amount of loading desired. Lossy media can be modeled by

introducing loss into the transmission line equations or by loading the nodes with lossy stubs.

Absorbing boundaries are easily constructed in TLM meshes by terminating each boundary node

transmission line with its characteristic impedance.

The advantages of using the TLM method are similar to those of the FDTD method. Complex,

nonlinear materials are readily modeled. Impulse responses and the time-domain behavior of

systems are determined explicitly. And, like FDTD, this technique is suitable for implementation

on massively parallel machines.

The disadvantages of the FDTD method are also shared by this technique. The primary

disadvantage being that voluminous problems that must use a fine grid require excessive amounts

of computation.

Nevertheless, both the TLM and FDTD techniques are very powerful and widely used. For

many types of EM problems they represent the only practical methods of analysis. Deciding

whether to utilize a TLM or FDTD technique is a largely personal decision. Many engineers find

the transmission line analogies of the TLM method to be more intuitive and easier to work with.

On the other hand, others prefer the FDTD method because of its simple, direct approach to the

V1

V2

V

V

V

V

V

V

V

V

VV

127

3

6

4

10

11

5

8

9

Figure 4: The Symmetrical Condensed Node

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solution of Maxwells field equations. The TLM method requires significantly more computer

memory per node, but it generally does a better job of modeling complex boundary geometries.

This is because both E and H are calculated at every boundary node.

A listing for a general purpose TLM code written in FORTRAN can be found in a Ph.D.dissertation by S. Akhtarzad [24]. This program can be adapted to a variety of applications. A

general overview of the TLM method and a two-dimensional TLM code is provided in a book by

Hoefer [25].

Generalized Multipole Technique

The Generalized Multipole Technique (GMT) is a relatively new method for analyzing EM

problems. It is a frequency domain technique that (like the method of moments) is based on the

method of weighted residuals. However, this method is unique in that the expansion functions are

analytic solutions of the fields generated by sources located some distance away from the surface

where the boundary condition is being enforced.

Moment methods generally employ expansion functions representing quantities such as charge

or current that exist on a boundary surface. The expansion functions of the Generalized Multipole

Technique are spherical wave field solutions corresponding to multipole sources. By locating these

sources away from the boundary, the field solutions form a smooth set of expansion functions on

the boundary and singularities on the boundary are avoided.

Like the method of moments, a system of linear equations is developed and then solved to

determine the coefficients of the expansion functions that yield the best solution. Since the

expansion functions are already field solutions, it is not necessary to do any further computation to

determine the fields. Conventional moment methods determine the currents and/or charges on the

surface first and then must integrate these quantities over the entire surface to determine the fields.

This integration is not necessary at any stage of the GMT solution.

There is little difference in the way dielectric and conducting boundaries are treated by the GMT.

The same multipole expansion functions are used. For this reason, a general purpose implementa-

tion of the GMT models configurations with multiple dielectrics and conductors much more readily

than a general purpose moment-method technique. On the other hand, moment method techniques,

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which employ expansion functions that are optimized for a particular type of configuration (e.g.

thin wires), are generally much more efficient at modeling that specific type of problem.

Over the last ten years, the GMT has been applied to a variety of EM configurations including

dielectric bodies [26,27], obstacles in waveguides [28], and scattering from perfect conductors[29,30]. Work in this young field is continuing and new developments are regularly announced.

Recent significant developments include the addition of a thin-wire modeling capability [31,32]

and a ringpole expansion function for modeling symmetric structures [33].

A commercial GMT code has been developed at the Swiss Federal Institute of Technology. This

code is called the MMP (Multiple MultiPole) code. A two-dimensional PC version is available

through Artech House Publishers [34]. A comprehensive text describing the GMT technique and

the MMP code is also available [35].

Conjugate Gradient Method

The conjugate gradient method is another technique based on the method of weighted residuals.

It is very similar conceptually to conventional moment method techniques. Nevertheless, there are

two features that generally distinguish this technique from other moment methods. The first has to

do with the way in which the weighting functions are utilized. The second involves the method of

solving the system of linear equations.

Conventional moment methods define the inner product of the weighting functions, wj, with

another function g as,

= s

(wjg) ds(14)

This is referred to as the symmetric product. The conjugate gradient method uses a different form

of the inner product called theHilbert inner product. This is defined as,

= s

(wjg) ds(15)

where the * denotes complex conjugation. If both functions are real, these two definitions are

equivalent. However, when complex weighting functions are utilized, the symmetric product is a

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complex quantity and therefore not a valid norm. In this case, the Hilbert inner product is preferred

[36].

The other major difference between conventional moment methods and the conjugate gradient

method involves the technique used to solve the large system of equations these methods generate.Conventional moment method techniques generally employ a Gauss-Jordan method or another

direct solution procedure. Direct solution techniques solve the system of equations with a given

number of calculations (generally O[N3], where N is the order of the matrix).

Conjugate gradient methods utilize an iterative solution procedure. This procedure, called the

method of conjugate gradients, can be applied to the system of equations or it can be applied directly

to the operator equation [37]. Iterative solution procedures such as the method of conjugate

gradients are most advantageous when applied to large, sparse matrices.

Boundary Element Method

The Boundary Element Method (BEM) is a weighted residual technique. It is essentially a

moment-method technique whose expansion and weighting functions are defined only on a

boundary surface. Most general purpose moment-method EM modeling codes employ a boundary

element method [13-16].

Like the finite element method, its origins are in the field of structural mechanics. Electrical

engineers are likely to use the more general term moment methodto describe an implementation of

this technique. Outside of electrical engineering however, the terms boundary element methodor

boundary integral element methodare commonly used.

Uniform Theory of Diffraction

The Uniform Theory of Diffraction (UTD) is an extension of the Geometrical Theory of

Diffraction (GTD). Both of these techniques are high-frequency methods. They are only accurate

when the dimensions of objects being analyzed are large relative to the wavelength of the field. In

general, as the wavelengths of an electromagnetic excitation approach zero, the fields can be

determined using geometric optics. UTD and GTD are extensions of geometric optics that include

the effects of diffraction.

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Diffraction is a local phenomena at high frequencies. Therefore, the behavior of the diffracted

wave at edges, corners, and surfaces can be determined from an asymptotic form of the exact solution

for simpler canonical problems. For example, the diffraction around a sharp edge is found by

considering the asymptotic form of the solution for an infinite wedge. GTD and UTD methods add

diffracted rays to geometric optical rays to obtain an improved estimate of the exact field solution.

The Basic Scattering Code (BSC) is a popular implementation of UTD. It is available from the

ElectroScience Laboratory of the Ohio State University [38].

Hybrid Techniques

It is apparent from the previous sections that none of the techniques described is well-suited to

all (or even most) electromagnetic modeling problems. Most moment method codes wont model

inhomogeneous, nonlinear dielectrics. Finite element codes cant efficiently model large radiation

problems. GMT and UTD codes are not appropriate for small, complex geometries or problems

that require accurate determination of the surface and wire currents. Unfortunately, most practical

printed circuit card radiation models have all of these features and therefore cannot be analyzed by

any of these techniques.

One solution, which has been employed by a number of researchers, is to combine two or more

techniques into a single code. Each technique is applied to the region of the problem for which it

is best suited. The appropriate boundary conditions are enforced at the interfaces between theseregions. Normally a surface integral technique such as the boundary element method will be

combined with a finite method such as the finite element, FDTD, or TLM method. Several

successful implementations of hybrid techniques are described in the literature [39-48].

So far, none of the available hybrid techniques model the radiation from printed circuit cards

very well. This is due to the fact that most of these methods were developed to predict radar cross

section (RCS) values or for other scattering problems where the source is remote from the

configuration being modeled. Work in this area is continuing however. Several researchers areinvolved in efforts to develop hybrid techniques that can be applied to a variety of presently

intractable problems.

Advances in the development and implementation of codes based on a single technique continue

to be important. However, there will always be problems that defy analysis by any one technique.

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Hybrid methods permit numerical modeling techniques to be applied to a whole new class of

configurations.

Conclusions

Several numerical modeling techniques have been described. A fundamental description of each

technique and an overview of the types of problems they are best suited to analyze have been

presented. References have been provided that direct the reader to more detailed information and

sources of computer codes.

The state-of-the-art in numerical modeling is progressing rapidly. Each year new types of

problems can be analyzed. Implementations of these techniques are getting more accurate and

powerful. Many practical EMC problems are already being solved using numerical computer

models. Before long, numerical modeling techniques are likely to become as indispensable to the

EMC engineer as they already are to the antenna and microwave engineer.

References

[1] J. L. Bogdanor, R. A. Pearlman, and M. D. Siegel, Intrasystem ElectromagneticCompatibility Analysis Program, vol. I-IV, McDonnell Douglas Aircraft Corp.,F30602-72-C-0277, Rome Air Development Center, Griffiss AFB, NY, Dec. 1974.

[2] C. R. Paul and D. R. Bush, Radiated Emissions from Common-Mode Currents, Proc. 1987IEEE Symp. EMC, Atlanta, GA, August, 1987, pp. 197-203.

[3] T. H. Hubing and J. F. Kauffman, Modeling the Electromagnetic Radiation from ElectricallySmall Table-Top Products,IEEE Trans. Electromag. Compat., vol. EMC-31, Feb 1989,pp. 74-84.

[4] D. R. Lynch and K. D. Paulsen, Origin of Vector Parasites in Numerical MaxwellSolutions,IEEE Trans. Microwave Theory and Tech., vol. MTT-39, March 1991,pp. 383-394.

[5] K. D. Paulsen and D. R. Lynch, Elimination of Vector Parasites in Finite Element MaxwellSolutions,IEEE Trans. Microwave Theory and Tech., vol. MTT-39, March 1991,pp. 395-404.

[6] MSC/EMASTM

, Finite-element software available from MacNeal-Schwendler Corporation,9076 North Deerbrook Trail, Milwaukee, WI 53223-2434

[7] MAXWELLTM

, Finite-element software available from Ansoft Corporation, 4516 HenryStreet, Pittsburgh, PA 15213

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[8] G. L. Maile, Three-Dimensional Analysis of Electromagnetic Problems by Finite ElementMethods, Ph.D. dissertation, University of Cambridge, December 1979.

[9] J. P. Webb, Developments in a Finite Element Method for Three-DimensionalElectromagnetic Problems, Ph.D. dissertation, University of Cambridge, September

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